The direct limit, also called a colimit, of a family of -modules is the dual notion of an inverse
limit and is characterized by the following mapping property. For a directed
set
and a family of -modules
, let be a direct system. is some -module with some homomorphisms , where for each , ,
(1)
such that if there exists some -module with homomorphisms , where for each , ,
(2)
then a unique homomorphism is induced and the above diagram
commutes.
The direct limit can be constructed as follows. For a given direct system, ,
(3)
letting
be the -module
generated by
where
and and are the images of and in .
-modules is the dual notion of an inverse
limit and is characterized by the following mapping property. For a directed
set 
and a family of 
-modules
, let 
be a direct system. 
is some 
-module with some homomorphisms 
, where for each 
, 
,
-module 
with homomorphisms 
, where for each 
, 
,
is induced and the above diagram
commutes.
,
be the 
-module
generated by 
where 
and 
and 
are the images of 
and 
in 
.
